Functional Graphical Models

ABSTRACT

Graphical models have attracted increasing attention in recent years, especially in settings involving high-dimensional data. In particular, Gaussian graphical models are used to model the conditional dependence structure among multiple Gaussian random variables. As a result of its computational efficiency, the graphical lasso (glasso) has become one of the most popular approaches for fitting high-dimensional graphical models. In this paper, we extend the graphical models concept to model the conditional dependence structure among p random functions. In this setting, not only is p large, but each function is itself a high-dimensional object, posing an additional level of statistical and computational complexity. We develop an extension of the glasso criterion (fglasso), which estimates the functional graphical model by imposing a block sparsity constraint on the precision matrix, via a group lasso penalty. The fglasso criterion can be optimized using an efficient block coordinate descent algorithm. We establish the concentration inequalities of the estimates, which guarantee the desirable graph support recovery property, that is, with probability tending to one, the fglasso will correctly identify the true conditional dependence structure. Finally, we show that the fglasso significantly outperforms possible competing methods through both simulations and an analysis of a real-world electroencephalography dataset comparing alcoholic and nonalcoholic patients.

Keywords:

• Block coordinate descent algorithm
• Block sparse precision matrix
• Functional data
• Functional principal component analysis
• Graphical models

Acknowledgment

The authors thank the Editor, the Associate Editor, and three referees for their useful comments and suggestions.

Notes

1 Here, we assume the same time domain, $\mathcal{T}$, for all random functions to simplify the notation, but our methodological and theoretical results extend naturally to the more general case where each function corresponds to a different time domain.

2 Here, we can use the projection theorem for Hilbert spaces (Hsing and Eubank 2015, chap. 2.5) to rigorously define the relevant conditional expectation terms, for example, E(g_j(s)|{g_k( · ), k ≠ j, l}). See also the definition of the conditional joint probability measure within Hilbert spaces in Zhu, Strawn, and Dunson (2016).

3 Our methodological and theoretical results can be extended to the case of Gaussian processes with nonzero means but for clarity of the exposition, we do not investigate that case here.

Additional information

Funding

Shaojun Guo was partially supported by the National Natural Science Foundation of China (No. 11771447), the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China.